Remarks on proper conflict-free colorings of graphs
A vertex coloring of a graph is said to be conflict-free with respect to neighborhoods if for every non-isolated vertex there is a color appearing exactly once in its (open) neighborhood. As defined in [Fabrici et al., Proper Conflict-free and Unique-maximum Colorings of Planar Graphs with Respect t...
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| Published in | Discrete mathematics Vol. 346; no. 2; p. 113221 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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01.02.2023
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| Abstract | A vertex coloring of a graph is said to be conflict-free with respect to neighborhoods if for every non-isolated vertex there is a color appearing exactly once in its (open) neighborhood. As defined in [Fabrici et al., Proper Conflict-free and Unique-maximum Colorings of Planar Graphs with Respect to Neighborhoods, arXiv preprint], the minimum number of colors in any such proper coloring of graph G is the PCF chromatic number of G, denoted χpcf(G). In this paper, we determine the value of this graph parameter for several basic graph classes including trees, cycles, hypercubes and subdivisions of complete graphs. We also give upper bounds on χpcf(G) in terms of other graph parameters. In particular, we show that χpcf(G)≤5Δ(G)/2 and characterize equality. Several sufficient conditions for PCF k-colorability of graphs are established for 4≤k≤6. The paper concludes with few open problems. |
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| AbstractList | A vertex coloring of a graph is said to be conflict-free with respect to neighborhoods if for every non-isolated vertex there is a color appearing exactly once in its (open) neighborhood. As defined in [Fabrici et al., Proper Conflict-free and Unique-maximum Colorings of Planar Graphs with Respect to Neighborhoods, arXiv preprint], the minimum number of colors in any such proper coloring of graph G is the PCF chromatic number of G, denoted χpcf(G). In this paper, we determine the value of this graph parameter for several basic graph classes including trees, cycles, hypercubes and subdivisions of complete graphs. We also give upper bounds on χpcf(G) in terms of other graph parameters. In particular, we show that χpcf(G)≤5Δ(G)/2 and characterize equality. Several sufficient conditions for PCF k-colorability of graphs are established for 4≤k≤6. The paper concludes with few open problems. |
| ArticleNumber | 113221 |
| Author | Škrekovski, Riste Caro, Yair Petruševski, Mirko |
| Author_xml | – sequence: 1 givenname: Yair surname: Caro fullname: Caro, Yair email: yacaro@kvgeva.org.il organization: University of Haifa-Oranim, Israel – sequence: 2 givenname: Mirko surname: Petruševski fullname: Petruševski, Mirko email: mirko.petrushevski@gmail.com organization: Department of Mathematics and Informatics, Faculty of Mechanical Engineering, Skopje, The Former Yugoslav Republic of Macedonia – sequence: 3 givenname: Riste surname: Škrekovski fullname: Škrekovski, Riste email: skrekovski@gmail.com organization: FMF, University of Ljubljana & Faculty of Information Studies, Novo mesto, Slovenia |
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| Keywords | Proper coloring PCF chromatic number Planar graph Conflict-free coloring Neighborhood |
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| SubjectTerms | Conflict-free coloring Neighborhood PCF chromatic number Planar graph Proper coloring |
| Title | Remarks on proper conflict-free colorings of graphs |
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